Supergravity in Two Spacetime Dimensions
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Selected Papers on Teleparallelism Ii
SELECTED PAPERS ON TELEPARALLELISM Edited and translated by D. H. Delphenich Table of contents Page Introduction ……………………………………………………………………… 1 1. The unification of gravitation and electromagnetism 1 2. The geometry of parallelizable manifold 7 3. The field equations 20 4. The topology of parallelizability 24 5. Teleparallelism and the Dirac equation 28 6. Singular teleparallelism 29 References ……………………………………………………………………….. 33 Translations and time line 1928: A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism ,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 217- 221………………………………………………………………………………. 35 (Received on June 7) A. Einstein, “A new possibility for a unified field theory of gravitation and electromagnetism” Sitzber. Preuss. Akad. Wiss. 17 (1928), 224-227………… 42 (Received on June 14) R. Weitzenböck, “Differential invariants in EINSTEIN’s theory of teleparallelism,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 466-474……………… 46 (Received on Oct 18) 1929: E. Bortolotti , “ Stars of congruences and absolute parallelism: Geometric basis for a recent theory of Einstein ,” Rend. Reale Acc. dei Lincei 9 (1929), 530- 538...…………………………………………………………………………….. 56 R. Zaycoff, “On the foundations of a new field theory of A. Einstein,” Zeit. Phys. 53 (1929), 719-728…………………………………………………............ 64 (Received on January 13) Hans Reichenbach, “On the classification of the new Einstein Ansatz on gravitation and electricity,” Zeit. Phys. 53 (1929), 683-689…………………….. 76 (Received on January 22) Selected papers on teleparallelism ii A. Einstein, “On unified field theory,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 2-7……………………………………………………………………………….. 82 (Received on Jan 30) R. Zaycoff, “On the foundations of a new field theory of A. Einstein; (Second part),” Zeit. Phys. 54 (1929), 590-593…………………………………………… 89 (Received on March 4) R. -
Book of Abstracts Ii Contents
Spring workshop on gravity and cosmology Monday, 25 May 2020 - Friday, 29 May 2020 Jagiellonian University Book of Abstracts ii Contents Stochastic gravitational waves from inflaton decays ..................... 1 Approximate Killing symmetries in non-perturbative quantum gravity .......... 1 Disformal transformations in modified teleparallelism .................... 1 Quantum fate of the BKL scenario ............................... 2 Ongoing Efforts to Constrain Lorentz Symmetry Violation in Gravity ........... 2 Proof of the quantum null energy condition for free fermionic field theories . 2 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 3 Stable Cosmology in Generalised Massive Gravity ...................... 3 Constraining the inflationary field content .......................... 3 The (glorious) past, (exciting) present and (foreseeable) future of gravitational wave detec- tors. ............................................. 4 reception ............................................. 4 Testing Inflation with Primordial Messengers ........................ 4 Gravitational waves from inflation ............................... 4 CDT, a theory of quantum geometry. ............................. 4 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 5 Loop-based observables in 4D CDT .............................. 5 Spectral Analysis of Causal Dynamical Triangulations via Finite Element Methods . 5 A consistent theory of D -> 4 Einstein-Gauss-Bonnet gravity ................ 5 Quantum Ostrogradsky theorem -
SUPERSYMMETRY 1. Introduction the Purpose
SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1. -
Proofs JHEP 092P 1215 Xxxxxxx ???, 2016 Springer : Doi: March 1, 2016 : Y February 25, 2016 December 11, 2015 : : Setting
Published for SISSA by Springer Received: December 11, 2015 Revised: February 25, 2016 Accepted: March 1, 2016 Published: ???, 2016 proofs JHEP_092P_1215 Notes on super Killing tensors P.S. Howea and U. Lindstr¨omb,c aDepartment of Mathematics, King’s College London, The Strand, London WC2R 2LS, U.K. bDepartment of Physics and Astronomy, Theoretical Physics, Uppsala University, SE-751 20 Uppsala, Sweden cTheoretical Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, U.K. E-mail: [email protected], [email protected] Abstract: The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the even Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed. Keywords: Extended Supersymmetry, Superspaces, Higher Spin Symmetry ArXiv ePrint: 1511.04575 Open Access, c The Authors. doi:xxxxxxx Article funded by SCOAP3. Contents 1 Introduction 1 2 Killing tensors in superspace 3 3 Superparticles 6 4 SCKTs in flat superspaces 10 proofs JHEP_092P_1215 5 Analytic superspace 18 6 Components of superconformal Killing tensors 24 6.1 D = 3 25 6.2 D = 4 26 6.3 D = 6 27 6.4 Algebras 28 7 Comments on higher spin 29 8 Concluding remarks 30 A Supersymmetric SN bracket 31 1 Introduction In ordinary Lorentzian spacetime a Killing vector, K, is a symmetry of the metric, i.e. -
Alternative Gravitational Theories in Four Dimensions
Alternative Gravitational Theories in Four Dimensionsa Friedrich W. Hehl Institute for Theoretical Physics, University of Cologne, D-50923 K¨oln, Germany email: [email protected] We argue that from the point of view of gauge theory and of an appropriate interpretation of the interferometer experiments with matter waves in a gravitational field, the Einstein- Cartan theory is the best theory of gravity available. Alternative viable theories are general relativity and a certain teleparallelism model. Objections of Ohanian & Ruffini against the Einstein-Cartan theory are discussed. Subsequently we list the papers which were read at the ‘Alternative 4D Session’ and try to order them, at least partially, in the light of the structures discussed. 1 The best alternative theory? I would call general relativity theory8 GR the best available alternative gravitational 21,14 theory and the next best one its teleparallel equivalent GR||. Because of these two theories, at least, it is good to have this alternative session during the Marcel Grossmann Meeting. Let me try to explain why I grant to GR the distinction of being the best alternative theory. After finally having set up special relativity theory in 1905, Einstein subse- quently addressed the question of how to generalize Newton’s gravitational theory as to make it consistent with special relativity, that is, how to reformulate it in a Poincar´ecovariant way. Newton’s theory was a battle tested theory in the realm of our planetary system and under normal laboratory conditions. It has predictive power as it had been shown by the prediction of the existence of the planet Neptune in the last century. -
Teleparallelism Arxiv:1506.03654V1 [Physics.Pop-Ph] 11 Jun 2015
Teleparallelism A New Way to Think the Gravitational Interactiony At the time it celebrates one century of existence, general relativity | Einstein's theory for gravitation | is given a companion theory: the so-called teleparallel gravity, or teleparallelism for short. This new theory is fully equivalent to general relativity in what concerns physical results, but is deeply different from the con- ceptual point of view. Its characteristics make of teleparallel gravity an appealing theory, which provides an entirely new way to think the gravitational interaction. R. Aldrovandi and J. G. Pereira Instituto de F´ısica Te´orica Universidade Estadual Paulista S~aoPaulo, Brazil arXiv:1506.03654v1 [physics.pop-ph] 11 Jun 2015 y English translation of the Portuguese version published in Ci^enciaHoje 55 (326), 32 (2015). 1 Gravitation is universal One of the most intriguing properties of the gravitational interaction is its universality. This hallmark states that all particles of nature feel gravity the same, independently of their masses and of the matter they are constituted. If the initial conditions of a motion are the same, all particles will follow the same trajectory when submitted to a gravitational field. The origin of universality is related to the concept of mass. In principle, there should exist two different kinds of mass: the inertial mass mi and the gravitational mass mg. The inertial mass would describe the resistance a particle shows whenever one attempts to change its state of motion, whereas the gravitational mass would describe how a particle reacts to the presence of a gravitational field. Particles with different relations mg=mi, therefore, should feel gravity differently when submitted to a given gravitational field. -
Supersymmetric Lagrangians
1 Section 1 Setup Supersymmetric Lagrangians Chris Elliott February 6, 2013 1 Setup Let's recall some notions from classical Lagrangian field theory. Let M be an oriented supermanifold: our spacetime, of dimension njd. For our purposes, we will most often think about the case where M is either Minkowski space Mˇ n = R1;n−1, or Super Minkowski space M = Mˇ n × ΠS where S is a representation of Spin(1; n − 1). Definition 1.1. Let E ! M be a smooth (super) fibre bundle. The associated space of fields is the space of smooth sections φM ! E.A Lagrangian density on F is a function L: F! Dens(M) { where Dens(M) denotes the bundle of densities on M { that satisfies a locality condition. Precisely, we form the pullback bundle Dens(M)F / Dens(M) M × F / M π1 and require L to be a section of this bundle such that, for some k, for all m 2 M, L(m; φ) only depends on the first k derivatives of φ. We can phrase this in terms of factoring through the kth jet bundle of E. As usual, we define the action functional to be the integral Z S(φ) = L(φ): M A Lagrangian system generally includes additional information, namely a choice of variational 1-form γ. In general I won't use this data, but I should mention when it might play a role. 1.1 Symmetries In this talk, we will be discussing Lagrangian systems with certain kinds of symmetry called supersymmetry. As such, it'll be important to understand what it actually means to be a symmetry of such a system. -
[Math.RA] 14 Jun 2007 the Minkowski and Conformal Superspaces
IFIC/06-28 FTUV-05/0928 The Minkowski and conformal superspaces R. Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5. 40126 Bologna. Italy. e-mail: fi[email protected] M. A. Lled´o Departament de F´ısica Te`orica, Universitat de Val`encia and IFIC C/Dr. Moliner, 50, E-46100 Burjassot (Val`encia), Spain. e-mail: Maria.Lledo@ific.uv.es and V. S. Varadarajan Department of Mathematics, UCLA. Los Angeles, CA, 90095-1555, USA e-mail: [email protected] Abstract We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it arXiv:math/0609813v2 [math.RA] 14 Jun 2007 as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra. 1Investigation supported by the University of Bologna, funds for selected research top- ics. 1 Contents 1 Introduction 3 2 The complex Minkowski space time and the Grassmannian G(2,4) 6 2.1 The Pl¨ucker embedding and the Klein quadric . 7 2.2 Therealform ........................... 10 3 Homogeneous spaces for Lie supergroups 11 3.1 The supermanifold structure on X = G/H ........... 12 3.2 The action of G on X ...................... 15 3.3 The functor of points of X = G/H ............... 16 4 The super Poincar´eand the translation superalgebras 17 4.1 Thetranslationsuperalgebra. -
Lectures on Supersymmetry and Superstrings 1 General Superalgebra
Lectures on supersymmetry and superstrings Thomas Mohaupt Abstract: These are informal notes on some mathematical aspects of su- persymmetry, based on two ‘Mathematics and Theoretical Physics Meetings’ in the Autumn Term 2009.1 First super vector spaces, Lie superalgebras and su- percommutative associative superalgebras are briefly introduced together with superfunctions and superspaces. After a review of the Poincar´eLie algebra we discuss Poincar´eLie superalgebras and introduce Minkowski superspaces. As a minimal example of a supersymmetric model we discuss the free scalar superfield in two-dimensional N = (1, 1) supersymmetry (this is more or less copied from [1]). We briefly discuss the concept of chiral supersymmetry in two dimensions. None of the material is original. We give references for further reading. Some ‘bonus material’ on (super-)conformal field theories, (super-)string theories and symmetric spaces is included. These are topics which will play a role in future meetings, or which came up briefly during discussions. 1 General superalgebra Definition: A super vector space V is a vector space with a decomposition V = V V 0 ⊕ 1 and a map (parity map) ˜ : (V V ) 0 Z · 0 ∪ 1 −{ } → 2 which assigns even or odd parity to every homogeneous element. Notation: a V a˜ =0 , ‘even’ , b V ˜b = 1 ‘odd’ . ∈ 0 → ∈ 1 → Definition: A Lie superalgebra g (also called super Lie algebra) is a super vector space g = g g 0 ⊕ 1 together with a bracket respecting the grading: [gi , gj] gi j ⊂ + (addition of index mod 2), which is required to be Z2 graded anti-symmetric: ˜ [a,b]= ( 1)a˜b[b,a] − − (multiplication of parity mod 2), and to satisfy a Z2 graded Jacobi identity: ˜ [a, [b,c]] = [[a,b],c] + ( 1)a˜b[b, [a,c]] . -
A Teleparallel Effective Geometry for Einstein's Unified Field Theory
A teleparallel effective geometry for Einstein’s unified field theory L.C. Garcia de Andrade 1 Abstract Riemannian and teleparallel geometrical approaches to the investi- gation of Maxwell electrodynamics shown that a unified field theory of gravitation and electromagnetism a la Einstein can be obtained from a stationary metric. This idea contrasts with the recently proposed pre- metric electrodynamics by Hehl and Obukhov. In the teleparallel case the definition of the electric field is obtained straightforward from the spacetime metric and the orthonormal basis frame of teleparallelism. In this case the only nonvanishing component of Cartan torsion is defined as the effective electric field. In this approach the gravita- tional potentials or metric coefficients are expressed in terms of the effective or analogous electric and magnetic potentials. Thefore the arXiv:gr-qc/0501050v1 16 Jan 2005 Maxwell equations in vacuum can be obtained by derivation of this electric field definition as usual. In the Riemannian case we consider an electrostatic spacetime where the Einstein equations in vacuum in the approximation of linear fields. The constraint of Einstein equa- tions in vacuum are shown to lead or to the Coulomb equation or to a singular behaviour on the metric which would represent a kind of effective electrodynamic black hole event horizon. 1Departamento de F´ısica Te´orica - Instituto de F´ısica Rua S˜ao Fco. Xavier 524, Rio de Janeiro, RJ Maracan˜a, CEP:20550-003 , Brasil. E-Mail.: [email protected] 1 Introduction Recently Hehl and Obukhov (HO) [1] have proposed a pre-metric electrody- namics where the metric would be oobtained from constitutive relations of electrodynamics. -
M-Theory from the Superpoint
M-Theory from the Superpoint John Huerta∗, Urs Schreibery March 21, 2017 Abstract The \brane scan" classifies consistent Green{Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The \brane bouquet" generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5- brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive maximal invariant central extensions, and show that it discovers the super- Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet. Contents 1 Introduction 2 2 Automorphisms of super-Minkowski spacetimes9 3 The consecutive maximal invariant central extensions of the superpoint 12 4 Outlook 17 A Background 17 A.1 Super Lie algebra cohomology............................... 17 A.2 Super Minkowski spacetimes................................ 20 ∗CAMGSD, Instituto Superior T´ecnico,Av. Ravisco Pais, 1049-001 Lisboa, Portugal yMathematics Institute of the Academy, Zitnaˇ 25, 115 67 Praha 1, Czech Republic, on leave at Bonn 1 1 Introduction In his \vision talk" at the annual string theory conference in 2014, Greg Moore highlighted the following open question in string theory [41, section 9]: Perhaps we need to understand the nature of time itself better. -
Dynamics in Interacting Scalar-Torsion Cosmology
universe Article Dynamics in Interacting Scalar-Torsion Cosmology Andronikos Paliathanasis 1,2 1 Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa; [email protected] 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile Abstract: In a spatially flat Friedmann–Lemaître–Robertson–Walker background space, we con- sider a scalar-torsion gravitational model which has similar properties to the dilaton theory. This teleparallel model is invariant under a discrete transformation similar to the Gasperini–Veneziano duality transformation. Moreover, in the gravitational action integral, we introduce the Lagrangian function of a pressureless fluid source which is coupled to the teleparallel dilaton field. This specific gravitational theory with interaction in the dark sector of the universe was investigated by using methods of the dynamical system analysis. We calculate that the theory provides various areas of special interest for the evolution of the cosmological history. Inflationary scaling solutions and the de Sitter universe are recovered. Furthermore, we calculate that there exist an attractor which provides a stable solution where the two fluid components, the scalar field and the pressureless matter, contribute in the cosmological fluid. This solution is of special interest because it can describe the present epoch. Finally, the qualitative evolution of the cosmographic parameters is discussed. Keywords: teleparallel; scalar field; dilaton field; scalar-torsion; interaction Citation: Paliathanasis, A. Dynamics in Interacting Scalar-Torsion 1. Introduction Cosmology. Universe 2021, 7, 244. The detailed analysis of the recent cosmological data indicates that General Relativity https://doi.org/10.3390/ may need to be modified in order to describe the observations; for a recent review, we refer universe7070244 the reader to [1].